The Darboux equation is a particular case of
the Heun equation which, in an algebraic form [1]}, [2]}, reads
\(\begin{array}{l}\frac{d^{2}H(x)}{dx^{2}}+\left[\frac{\gamma }{x}+\frac{\delta }{x-1}+\frac{\epsilon }{x-a}\right]\frac{dH(x)}{dx}+\frac{\alpha \beta x-q}{x(x-1)(x-a)} H(x)=0,\quad \epsilon =\alpha +\beta +1-\gamma -\delta ,\end{array}\) 
